On sharpness of tchebycheff-type inequalities

Summary In this paper we discuss the sharpness of Tchebycheff-type inequalities obtained by the standard method, and give a unified theory on sharp inequalities. The problem of existence of a probability distribution that attains the sharp bound is also considered. The proof, based on the theory of convex sets, is very simple and the results might be sufficiently general for practical applications.     

Determining lower and upper bounds on probabilities of atomic propositions in sets of logical formulas represented by digraphs

In this paper we consider the problem of determining lower and upper bounds on probabilities of atomic propositions in sets of logical formulas represented by digraphs. We establish a sharp upper bound, as well as a lower bound that is not in general sharp. We show further that under a certain condition the lower bound is sharp. In that case, we obtain a closed form solution for the possible probabilities of the atomic propositions.The second author is partially supported by ONR grant N00014-92-J-1028 and AFOSR grant 91-0287.     

Weak Sharp Solutions for Nonsmooth Variational Inequalities

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set.     

Weak sharp solutions for generalized variational inequalities

We consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued, and not necessarily monotone. We extend the concept of weak sharpness to this more general framework, and establish some of its characterizations. We establish connections between weak sharpness and (1) gap functions for variational inequalities, and (2) global error bound. When the solution set is weak sharp, we prove finite convergence of the sequence generated by an arbitrary algorithm, for the monotone set-valued case, as well as for the case in which the underlying set-valued map is either Lipschitz continuous in the set-valued sense, for infinite dimensional spaces, or inner-semicontinuous when the space is finite dimensional.     

Morse inequalities for the Koszul complex of multi-persistence

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.     

On the stable containment of two sets

This paper studies the stability of the set containment problem. Given two non-empty sets in the Euclidean space which are the solution sets of two systems of (possibly infinite) inequalities, the Farkas type results allow to decide whether one of the two sets is contained or not in the other one (which constitutes the so-called containment problem). In those situations where the data (i.e., the constraints) can be affected by some kind of perturbations, the problem consists of determining whether the relative position of the two sets is preserved by sufficiently small perturbations or not. This paper deals with this stability problem as a particular case of the maintaining of the relative position of the images of two set-valued mappings; first for general set-valued mappings and second for solution sets mappings of convex and linear systems. Thus the results in this paper could be useful in the postoptimal analysis of optimization problems with inclusion constraints.      

Mixed integer formulations for the probabilistic minimum energy broadcast problem in wireless networks

In this paper we study a new variant of the minimum energy broadcast (MEB) problem, namely the probabilistic MEB (PMEB). The objective of the classic MEB problem is to assign transmission powers to the nodes of a wireless network is such a way that the total energy dissipated on the network is minimized, while a connected broadcasting structure is guaranteed by the assigned transmission powers. In the new variant of the problem treated in this paper, node failure is taken into account, aiming at providing solutions with a chosen reliability level for the broadcasting structure. Three mixed integer linear programming formulations for the new problem are presented, together with efficient formulation-dependent methods for their solution. Computational results are proposed and discussed. One method emerges as the most promising one under realistic settings. It is able to handle problems with up to fifty nodes.     

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n ? 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem’s coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.     

On Some New Integral Inequalities for Functions in One and Two Variables

In this paper, we consider a bound on a general version of the integral inequalities for functions and also study the qualitative behavior of the solutions of certain classes of the hyperbolic partial delay differential equations under the integral inequalities.     

On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula for the mode of the generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.     

Inequalities for the capacity of non-contractible annuli on cylinders of constant and variable negative curvature

We give elementary estimates for the capacity of non-contractible annuli on cylinders and provide examples, where these inequalities are sharp. Here the lower bound depends only on the area of the annulus. To obtain this result we use projection of gradients on curves to obtain a lower bound on the capacity, which we call directional capacity. In the case of constant curvature we then apply a symmetrization process that results in an annulus of minimal directional capacity. For this annulus the lower bound on the capacity is sharp. In the case of variable negative curvature we obtain the lower bound by constructing a comparison annulus with the same area but lower directional capacity on a cylinder of constant curvature. The methods developed here have been applied to estimate the energy of harmonic forms on Riemann surfaces in Muetzel (Math Zeitschrift, 2012, arXiv:1202.0782).     

Continuous dependence on the time and spatial geometry for the navier-stokes equations

In this paper we derive explicit a priori inequalities which imply stability of the solution of the initial-boundary value problem for the Navier-Stokes equations under perturbations of the initial time geometry and of the spatial geometry. These inequalities bound the solution perturbation In L₂ in terms of some well defined measure of the perturbation in geometry. We establish continuous dependence on spatial geometry in both two and three dimensions and continuous dependence on initial geometry in two dimensions. In the latter problem the three dimensional case will be somewhat more complicated due to the different form of the Sobolev inequality.     

Approximating schedules for dynamic process graphs efficiently

A model for parallel and distributed programs, the dynamic process graph (DPG), is investigated under graph-theoretic and complexity aspects. Such graphs embed constructors for parallel programs, synchronization mechanisms as well as conditional branches. They are capable of representing all possible executions of a parallel or distributed program in a very compact way. The size of this representation can be as small as logarithmic with respect to the size of any execution of the program.

In a preceding paper [A. Jakoby, et al., Scheduling dynamic graphs, in: Proc. 16th Symposium on Theoretical Aspects in Computer Science STACS'99, LNCS, vol. 1563, Springer, 1999, pp. 383–392] we have analysed the expressive power of the general model and various variants of it. We have considered the scheduling problem for DPGs given enough parallelism taking into account communication delays between processors when exchanging data. Given a DPG the question arises whether it can be executed (that means whether the corresponding parallel program has been specified correctly), and what is its minimum schedule length.

In this paper we study a subclass of dynamic process graphs called -output DPGs, which are appropriate in many situations, and investigate their expressive power. In a previous paper we have shown that the problem to determine the minimum schedule length is still intractable for this subclass, namely this problem is -complete as is the general case. Here we will investigate structural properties of the executions of such graphs. A natural graph-theoretic conjecture that executions must always split into components that are isomorphic to subgraphs turns out to be wrong. We are able to prove a weaker property. This implies a quadratic upper bound on the schedule length that may be necessary in the worst case, in contrast to the general case, where the optimal schedule length may be exponential with respect to the size of the representing DPG. Making this bound constructive, we obtain an approximation to a -complete problem. Computing such a schedule and then executing the program can be done on a parallel machine in polynomial time in a highly distributive fashion.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser–Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser–Trudinger trace inequalities, sharp Adams/Moser–Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.     

关于一个单形及其子单形的内切球半径的一些几何不等式(英文)

In this paper, we obtain some geometric inequalities on the radii of inscribed sphere of a simplex and its subsimplex, as particular case of this paper, we obtain some main results of [1].     

A Unified Treatment of Half-Discrete Hilbert-Type Inequalities with a Homogeneous Kernel

In this paper we provide a unified treatment of half-discrete Hilbert-type inequalities with a general homogeneous kernel. The main results are proved for the case of non-conjugate exponents. A special emphasis is given to determining conditions under which these inequalities include the best possible constants. As an application, we consider some operator expressions closely connected to established inequalities. Finally, we also provide improvements of derived half-discrete inequalities by virtue of the Hermite-Hadamard inequality.     

Strong KKT conditions and weak sharp solutions in convex-composite optimization

Using variational analysis techniques, we study convex-composite optimization problems. In connection with such a problem, we introduce several new notions as variances of the classical KKT conditions. These notions are shown to be closely related to the notions of sharp or weak sharp solutions. As applications, we extend some results on metric regularity of inequalities from the convex case to the convex-composite case.     

Schwarz methods with local refinement for the p-version finite element method

Summary. In some applications, the accuracy of the numerical solution of an elliptic problem needs to be increased only in certain parts of the domain. In this paper, local refinement is introduced for an overlapping additive Schwarz algorithm for the $-version finite element method. Both uniform and variable degree refinements are considered. The resulting algorithm is highly parallel and scalable. In two and three dimensions, we prove an optimal bound for the condition number of the iteration operator under certain hypotheses on the refinement region. This bound is independent of the degree $, the number of subdomains $ and the mesh size $. In the general two dimensional case, we prove an almost optimal bound with polylogarithmic growth in $. Received February 20, 1993 / Revised version received January 20, 1994     

非紧集上的变分不等式与拟变分不等式

本文在非常一般的框架和较弱的条件下证明了一类变分不等式与拟变分不等式解的存在性，将［１－７］的结果作了推广并改进在非紧集上讨论